The theory of determinants in the historical order of development, by Sir Thomas Muir.

418 HISTORY OF THE THEORY OF DETERMINANTS

thence, by merely changing the sign of a and the signs of all the
b's, that
P- a b,(b2- a) -c-/3 F+.b,
2 + b2 + _ b3(b3- a) = -
2 3 - b2++ b3 b 2 F - b1
/3-b3~b4 +
bn(bn-a)
+ P - bn + a
and therefore, by multiplication, that the product of the two continuedfractions on the left is equal to
_2 _ a2
4
The paper concludes with two other curious theorems of similar
character.*
* The rather noteworthy theorem in continuants which is the basis of this
striking result in continued-fractions can be still more widely generalized as follows.
The continuant
— +A b,
S1
b2 s
cl + — A + slcI b.
S2 r1
b.3 sb.2
Ss r2
b t + sb A + s -c3.. C3 S +3sC3 8b4
S4 r3...4 A + s4c4
S5 r4
is unaltered in value by adding
rlCl, r2c2 - 1C1, r3c, - s2c2, r4C4 - 3C3, - s404
to the elements of the main diagonal, and changing the elements of the upper minor
diagonal into
rib,, 2 A b)A s +, 4b (l 3)A.
o2' s r 3
Putting in this rl, r2,... sl, s... we have the identity
b + A b *
s1
c + A + c, b2.
82
C2 — + A + sc2 63
c -- + A + s3c3 b4
84

C4 5 + A + S4C4
85